Fractional strain gradient plasticity and ductile fracture of metals

We present an optimal scaling analysis based on (possibly fractional) strain-gradient plasticity. The analysis yields optimal scaling laws, in the sense of upper and lower bounds of a power-law type with matching exponents, connecting macroscopic fracture properties, such as the critical elongation at failure and the specific fracture energy, to microscopic mechanisms such as cleavage and microplasticity. We show that an optimal upper bound can be derived from an exceedingly simple test deformation that opens up a sheet of parallelepipedic voids. We also show that an optimal lower bound can be obtained by relaxing compatibility between transverse fibers, which effectively renders the analysis one-dimensional. The analysis predicts a 'gating effect' of the surface energy. Specifically, a critical surface energy arises from the analysis that marks a sharp transition between brittle and ductile behavior. When the surface energy of the material exceeds the threshold value, the macroscopic specific fracture energy is predicted to rise sharply as a power of the surface energy with an exponent defined precisely by the theory.